Plancherel Densities on Yu-Induced Tempered Components

Definitions and mathematical statement

The supplied record does not specify enough data to state a uniquely determined Plancherel-measure identity. The following gives a conditional formulation of its intended target and identifies the missing qualifications.

In the usual nonarchimedean representation-theoretic setting, let $F$ be a nonarchimedean local field, let $G$ be a connected reductive group over $F$, and let $\widehat{G(F)}_{\mathrm{temp}}$ denote the set of equivalence classes of irreducible tempered representations of $G(F)$. For a parabolic subgroup $P=MN$, with Levi factor $M$ and unipotent radical $N$, write \[ I_P^G(\sigma)=\operatorname{Ind}_{P(F)}^{G(F)}\left(\delta_P^{1/2}\sigma\right) \] for normalized parabolic induction, where $\delta_P$ is the modulus character and $\sigma$ is inflated from $M(F)$ to $P(F)$. This specifies notation in a conventional setting; the record does not establish the intended range of fields and groups or the additional hypotheses required for Yu's construction.

Requested assertion, subject to the unresolved specifications below. For every nondiscrete component of $\widehat{G(F)}_{\mathrm{temp}}$ arising by parabolic induction from nonsingular Yu supercuspidal representations of Levi subgroups, verify the Plancherel-measure formula referred to in the source as the Hiraga--Ichino--Ikeda Plancherel-measure conjecture.

Here the terms "nonsingular Yu supercuspidal" and "nondiscrete component" retain their source-specific meanings. The supplied record gives neither their precise conventions nor the conjectured density formula. In particular, it supplies no parameterization of the relevant components, Haar-measure normalization, reference measure for a density, or complete hypotheses. Consequently, the requested assertion is a qualified target for formalization, rather than a fully specified measure identity; these omissions must be resolved from the cited source before a precise mathematical statement can be finalized.

Short English statement

The problem asks for the Hiraga--Ichino--Ikeda Plancherel-measure formula on the nondiscrete tempered components obtained by inducing nonsingular Yu supercuspidal representations. The supplied record omits the formula and essential hypotheses and normalizations, so a complete precise conjecture cannot yet be extracted. Its open status is supplied dated curation, not an independently verified conclusion.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30004797$, source problem code catalogue entry, title "Plancherel Densities on Yu-Induced Tempered Components," in the supplied Oberwolfach Reports Open Problems collection.
• Supplied primary-source citation: Automorphic Forms, Geometry and Arithmetic (2022), Oberwolfach Reports, DOI 10.4171/owr/2021/39, pp. 2089--2156. https://doi.org/10.4171/owr/2021/39. The record assigns proposed year 2021 and supplies no individual proposer or exact problem location within the report.
• Supplied literature source, labeled "formal degrees and parabolic induction": https://doi.org/10.1007/s00229-025-01632-z.
• Supplied literature source, labeled "affine Hecke algebras and HII Plancherel conjectures": https://arxiv.org/abs/1807.10232.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the target open and states that recent work addresses only narrower generic or otherwise special parabolic-induction cases. This is supplied dated curation derived from the OWR record, not independently verified literature assessment; the time-sensitive status and cited claims require independent verification.
• The supplied corrected_verified statement status is explained as a publication-format audit that preserved mathematical content while repairing notation or typography. It does not establish mathematical correctness or independent source verification.